Hopf monoids from class functions on unitriangular matrices

Abstract

We build, from the collection of all groups of unitriangular matrices, Hopf monoids in Joyal's category of species. Such structure is carried by the collection of class function spaces on those groups, and also by the collection of superclass function spaces, in the sense of Diaconis and Isaacs. Superclasses of unitriangular matrices admit a simple description from which we deduce a combinatorial model for the Hopf monoid of superclass functions, in terms of the Hadamard product of the Hopf monoids of linear orders and of set partitions. This implies a recent result relating the Hopf algebra of superclass functions on unitriangular matrices to symmetric functions in noncommuting variables. We determine the algebraic structure of the Hopf monoid: it is a free monoid in species, with the canonical Hopf structure. As an application, we derive certain estimates on the number of conjugacy classes of unitriangular matrices.

Final Version, 32 pages, accepted in "Algebra and Number Theory"

References in corpus (1)

• Supercharacter theories and Schur rings

Cited by in corpus (13)

• The Hopf monoid on nonnesting supercharacters of pattern groups
• Hopf Algebras and Markov Chains
• Antipode and Primitive elements in the Hopf Monoid of Super Characters
• Supercharacter theories constructed by the method of little groups
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• Strong forms of linearization for Hopf monoids in species
• The combinatorics of ${\mathrm{GL}}_n$ generalized Gelfand--Graev characters
• Cancelation free formula for the antipode of linearized Hopf monoid
• Closed expressions for averages of set partition statistics
• Strong forms of self-duality for Hopf monoids in species
• Supercharacter theories of type $A$ unipotent radicals and unipotent polytopes
• A unipotent realization of the chromatic quasisymmetric function
• Positive Self-Dual Hopf Algebras of Galois Characters